Transferring select files from Old iMac to new iMac

Are you able to enter Time Machine and browse the backups in your Time Mahine drive?

Note: you may have to hold down the Option key and select "Browse Other Backup Discs…"

Then you a pick individual files and/or folders to restore to your new Mac. They will go into the same location that they were on your old Mac.

As to the adaptor to use I'd take the TM drive into a store that carries adaptors and see witch one will work with the drive and a new iMac.

Migration Mac to Mac or using Time Machine is all or nothing.

One solution, is to copy your Music Library, Photos Library and Documents Folder to an external hard drive or flash drive.

Then Import the Music and Photos Libraries into those App's, and copy the Documents into the Documents Folder of the new Mac.

When you say “select files” that does not tell us very much, if you can be more descriptive that may help. If it’s just a few small files then there are lots of methods.

However for most it is wise to use Migration Assistant and connect the old Macs Time Machine External Hard Disk and migrate. Then once the migration is completed to move unwanted items to Trash and empty the trash.